Coleman's side beat Georgia 1-0 in Tbilisi to ensure they would be in pole position for second spot ahead of their group finale with the Republic of Ireland on Monday.
Lawrence was replaced by teenager Ben Woodburn in both of the last two games.
He said: "We are in with a great shout and the pressure and spotlight is welcomed by me, I promise you".
Serbia are still favourites to win the group with a final home game against Georgia on Monday, while Wales play third-placed Ireland in what looks like becoming a shoot-out to see who finishes as runners-up to claim a probable playoff spot.
Gareth Bale will miss Wales' last two World Cup qualifiers against Georgia and the Republic of Ireland after suffering a calf strain. Last month, Chris Coleman won his first double header of qualifiers in six years as Wales manager.More news: 'Calm Before the Storm': Trump Makes Cryptic Remarks at Military Dinner
Wales' last win without Bale came in Macedonia four years ago, with three draws and three defeats in his absence since then. He won't be thinking this is his 50th cap - it's about what we need to do and how we can implement it. We need the senior boys to take responsibility'.
"We didn't have that attitude maybe, myself included". And let me tell you, David Brooks is a player that you're going to notice very, very quickly. He, along with Joe Allen, Joe Ledley and other players, know what it's all about.
In the standing table of World Cup Qualifying UEFA, Wales is in the second place with 14 points and the Georgia is in the fifth place with five points.
"It would not be a surprise to me when we win both games".
Wales looked the more likely to add a second goal before eventually settling for a one-goal victory, with Georgia offering little as an attacking force despite holding the possession advantage.
"Obviously, you're missing a world-class player and the team's talisman, but sometimes that can bring the lads together and it won't be the first time they've coped without him".